Magnetic order in ferromagnetically coupled spin ladders

Magnetic order in ferromagnetically coupled spin ladders S. Dalosto and J. Riera

arXiv:cond-mat/9911271v2 [cond-mat.str-el] 30 Mar 2000

Instituto de F´ısica Rosario, Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, y Departamento de F´ısica, Universidad Nacional de Rosario, Avenida Pellegrini 250, 2000-Rosario, Argentina (July 13, 2011) A model of coupled antiferromagnetic spin-1/2 Heisenberg ladders is studied with numerical techniques. In the case of ferromagnetic interladder coupling we find that the dynamic and static structure factor has a peak at (π, π/2) where the first (second) direction is along (transversal) to the ladders. Besides, we suggest that the intensity of this peak and the spin-spin correlation at the maximum distance along the ladder direction remain finite in the bulk limit for strong enough interladder coupling. We discuss the relevance of these results for magnetic compounds containing ladders coupled in a trellis lattice and for the stripe scenario in high-Tc superconducting cuprates.

PACS: 75.10.Jm, 75.40.Mg, 74.72.-h, 75.50.Ee

possibility of using this powerful technique is severely reduced. In this sense, one of the objectives of the present work is to study a model in which the frustrated AF interladder couplings of the trellis lattice are replaced by much simpler ferromagnetic (FM) couplings in a square lattice. We expect that some of the physics of the frustrated system can be captured by this effective simplified model.7 Besides, there are compounds which consist of FM coupled ladders like SrCu2 O3 . The results of the present work could be relevant to other FM coupled gaped systems like the dimerized chains in (VO)2 P2 O7 .8 Previous studies have compared the behavior of AF and FM frustrated and nonfrustrated coupled gapless spin systems (spin chains).9 Alternatively, a renewed interest in coupled ladders comes from the high-Tc cuprate superconductors themselves. A number of recent experiments, mainly neutron scattering studies10 , indicate the presence of incommensurate spin correlations which in turn have been interpreted as coming from the segregation of charge carriers into 1D domain walls or “stripes” leaving the regions between them as undoped antiferromagnets. There are several theoretical scenarios that have predicted or that attempt to explain this stripe order11–13 but the origin of this order and its relation to superconductivity is still controversial. In particular, the problem of stripe formation in a 2D microscopic model like the t-J or Hubbard models is extremely difficult to study with analytical or numerical techniques. In principle, the inclusion of charge and spin degrees of freedom is essential for the understanding of this problem. However, it has been suggested that assuming the presence of a stripe structure it is very instructive to study its magnetic properties by using a model with spin degrees of freedom only.14,15 In this simplified model, the AF insulating regions between the charged stripes are considered as n-leg isotropic ladders coupled by an effective interaction. In one such study15 , following the initial picture from Ref. 10, the insulating regions were considered as 3-leg ladders coupled by AF interactions. However, a numerical study of the 2D t-J model16 , as well as early studies of charge in-

I. INTRODUCTION

One of the main topics in condensed matter physics in recent years has been the study of low-dimensional antiferromagnetic spin systems. The strong interest in this field has been sparkled by the realization that CuO2 planes play an essential role in high-Tc superconductors, which was followed by the appearance of several compounds characterized by the presence of strong electronic correlations. These compounds include many cuprates, nickelates, vanadates and manganites, and are characterized by important and unique properties. In most of them the proximity of low-dimensional antiferromagnetic (AF) phases are the key to understand these properties. At the same time, the concept of spin ladder1 , originally introduced to explain the presence of a spin gap in (VO)2 P2 O7 and later in layered cuprates like Srn−1 Cun+1 O2n (Refs. 2,3) became an important theoretical tool to understand the behavior of strongly correlated systems. The physics of the two-leg spin ladder is characterized by the existence of a singlet-triplet spin gap and an exponential decay of correlation functions. The ground state which can be thought to a good approximation as a product of singlets living on the rungs is now well understood. However, the above mentioned cuprates, the important case of Sr14 Cu24 O41 , as well as many other compounds like CaV2 O5 , actually contain layers of coupled two-leg ladders. These ladders are coupled by frustrated interactions in a trellis lattice which make its study with analytical or numerical techniques quite difficult. In principle, in the absence of frustration, a reduction of the gap as the interladder coupling increases is expected. Eventually, the system becomes gapless at a quantum critical point (QCP)3,4 and for larger coupling it behaves essentially as a two-dimensional (2D) spin-1/2 square antiferromagnet. Much less is known for the trellis lattice, although a Schwinger boson study5 suggests the transition from a spin liquid to a possible spiral order as the interladder coupling (ILC) increases. Quantum Monte Carlo (QMC) studies6 of this frustrated system are hampered by the “minus sign problem” and the 1

homogeneities in Hubbard and t-J models11 , indicate the formation of “bond-centered” stripes, i.e. doped two-leg ladders alternating with undoped two-leg ladders. Coupled spin two-leg ladders were also studied14 but its relevance to the physics of Cu-O planes is relative since they miss the essential ingredient that the magnetic order of the AF slices is π-phase shifted as emphasized in Ref. 16. Hence, the second motivation for the present work comes from the assumption that this π-phase-shift between ladders can be modeled by taking a ferromagnetic coupling between them. In summary, the purpose of the present work is to study ferromagnetically coupled two-leg ladders and compare their behavior with the case of AF coupling. If this model is considered as an approximation of AF systems in the trellis lattice, the FM coupling is an effective interaction coming from the frustrated interactions between ladders. If this model is considered as an approach to the stripe phase of the cuprates, the FM interaction comes from a collective effect determined from the competition of charge and spin degrees of freedom. In both cases, the results of the present study lead to predictions which can be tested experimentally. We use essentially numerical techniques like QMC (world-line algorithm) which allows us to reach low enough temperatures so as to capture ground state properties, and exact diagonalization with the Lanczos algorithm (LD), complemented by the continued fraction formalism to compute dynamical properties.

forming a two-leg ladder with FM leg and AF rung interactions. This second case has already been studied numerically18 but it is not relevant for the problems we wish to address. Besides, we consider the case of FMcoupled AF plaquettes instead of dimers in which case we have a two-leg ladder with FM-AF alternating interactions along the legs (Fig. 1(b)).

1 1

1

1

1

J inter

(a)

J inter

(b)

(c)

FIG. 1. (a) Coupled dimers forming an alternating chain; (b) coupled plaquettes forming an alternating ladder; (c) coupled ladders in a square lattice. Full lines (dashed) correspond to AF (FM) interactions.

Our first concern in this section is the behavior of the spin gap starting from the situation of isolated dimers or plaquettes. Using exact diagonalization we computed the spin gap ∆ by substracting the energies in the S = 0 and S = 1 subspaces on finite clusters with up to 24 sites. The extrapolation to the bulk limit was done using the law ∆∞ + b exp(−L/L0 )/L. The final result is shown in Fig. 2. We notice that in the limit Jinter → −∞ we

II. QUASI-ONE DIMENSIONAL STUDY.

To gain insight about the effects of FM interladder couplings we start from the case of FM coupled AF dimers which are the simplest systems with a spin gap. We are thus led to 1D or quasi-1D systems which are much easier to study from the numerical point of view. Besides, systems with a random distribution of AF and FM couplings have received some theoretical attention and their possible physical realization in SrCuPt1−p Irp O6 has been discussed.17 The Hamiltonian is given by:

1.0

0.8

0.6

H = Hdimer + Hinter

∆

where:

0.4

Hdimer = J

X

Sa;1 · Sa;2 ,

a

Hinter =

X

0.2

Jinter,a,b,i,j Sa;i · Sb;j ,

(1)

a,b,i,j

0.0 0.0

where a is a dimer index and i = 1, 2 labels the sites in a dimer. J = 1 for simplicity. Periodic boundary conditions are imposed in the longitudinal direction. There are several ways of coupling dimers. We consider here the simplest case of dimers forming a FM-AF alternating chain (Fig. 1(a)). Another possibility is that of dimers

1.0

2.0

3.0

1/λ FIG. 2. Singlet-triplet spin gap in the bulk limit of FM coupled dimers (circles) and plaquettes (triangles) as a function of λ = −Jinter . The cross indicates the spin gap for the spin-1 chain ∆ = 0.41 divided by 4 (from Ref. (19)).

2

On the other hand, the behavior of the coupled plaquettes system is not obviously predictable. The second point we want to examine is the behavior of the excitations of these systems, in particular the S = 1 excitations as can be measured by neutron scattering experiments. For this purpose, using conventional Lanczos techniques with the continued fractions formalism, we have computed the dynamical structure function (zzcomponent) S(q, ω)21 which is shown in Fig 3. Already for the simplest case of coupled dimers (Fig 3(a),(b)) one can see an interesting feature which we will observe also for the coupled ladders case in the next section. The position of the gap which is at q = π for AF dimerized chains shifts to π/2 for the case of FM coupled dimers.22 A similar behavior is observed for coupled plaquettes (Fig 3(c),(d)), where the lowest energy peak changes from (qx , qy ) = (π, π) to (π/2, π) (x is the longitudinal direction) by switching from AF to FM interplaquette couplings. In both cases, there is a transfer of spectral weight from the original AF peak to the FM one. This behavior is independent of the absolute value of Jinter , except for finite size effects.

recover the cases of a spin-1 chain for the coupled dimer system and a spin-1 ladder for the coupled plaquette one. It is easy to realize (by solving a two-dimer system and a two-site spin-1 system) that the gap for the spin-1 chain is four times larger than the gap obtained by the coupled dimer system when Jinter → −∞ and the gap for the spin-1 ladder is twice larger than the coupled plaquette system in this limit. Thus, in the former case we obtain a gap ∆cd ×4 = 0.410, coincident with the value already reported in the literature.19 For the spin-1 ladder we would obtain a gap ∆cp × 2 = 0.290 ± 0.008, smaller than the gap for the S=1 chain as predicted theoretically.20 Qualitatively, the important feature here is that the gap decreases monotonically from the isolated dimers (or plaquettes) case as Jinter → −∞. In the case of FM coupled dimers, a monotonic behavior could be guessed from the fact that this system continuously evolves towards the valence-bond-solid picture of a spin-1 chain in that limit. 8

8

(a)

S(q,ω) (a.u.)

6

Jinter=0.4

(b) 6

Jinter=−0.4 III. COUPLED LADDERS.

4

4

2

2

0

0

π/2

0.0

1.0

2.0

ω/J

3.0

0.0

H = Hladder + Hinter

π

1.0

2.0

ω/J

3.0

4.0

where: Hladder = J

160.0

160

120

Jinter=0.8

120.0

Hinter = Jinter

Jinter=−0.8

80.0

80

(0,π) (π/2,π) (π,π)

0.0

0

1.0

2.0

ω/J

3.0

0.0

1.0

2.0

ω/J

3.0

X

X

Sa;1,i · Sa;2,i ,

a,i

Sa;2,i · Sa+1;1,i ,

(2)

where a stands now for a ladder index and l = 1, 2 for the two legs in a ladder. We have taken the case of an isotropic ladder by simplicity. J is again taken as the unit of energy. We start with the study of the ground state energy of the system of both FM and AF coupled ladders. To this purpose we have performed standard QMC simulations for L × L lattices with L = 4, 6, 8, 12, and 16. Periodic boundary conditions in both directions are considered. For each lattice and set of coupling constants, we took T/J = 0.125, 0.100 and 0.07, and the Trotter number M at each temperature such that the error due to the time discretization is comparable or smaller than the statistical error. Typically, M = 140 for T/J = 0.07. We made runs up to 106 MC steps for both thermalization and measurement. We computed the energy in the total Sz = 0 and Sz = 1 subspaces (E0 and E1 respectively). For each set of coupling constants, we extrapolated the

(π,0)

0.0

Sa;l,i · Sa;l,i+1 + J

a,i

(π/2,0)

40.0

40

X a,l,i

(d)

(c) S(q,ω) (a.u.)

We have now arrived at the central part of this work. The Hamiltonian for the system we consider now (Fig. 1(c)) is essentially the same as (1) which we rewrite here for clarity:

4.0

FIG. 3. Dynamical structure factor as a function of the frequency for (a) AF and (b) FM coupled dimers for several momentum increasing from qx = π/10 to π in steps of π/10 from top to bottom. These results were obtained for the 20 site chain with LD. The same for (c) AF and (d) FM coupled plaquettes. In this case, q = (qx , qy ), where x is the longitudinal direction, are shown in steps of qx = π/4. These results were obtained for the 2x8 cluster with LD.

3

(0,1) correlation which increases faster in the latter. Of corresponding energies per site e0 and e1 to the bulk course, this correlation is negative (positive) in the AF limit using the law e∞ + b exp(−L/L0 )/L2 in the gaped (FM) case. region and e∞ + b/L3 in the gapless case. We obtained This indication of a difference between the two ILC very close values for e∞,0 and e∞,1 which gives an incases can be traced to a more intimate level which would dication of the good quality of the fits. The results for also provide experimentally measurable features. To this the ground state energy for FM and AF ladder couplings end, let us examine now the static structure factor S(q) are shown in Fig. 4. An interesting feature can be noobtained by Fourier transforming the spin-spin correlaticed: the energies for both signs of Jinter are degenerate tions obtained by QMC at the lowest temperature atwithin numerical errors for |Jinter | < 0.3. This situatained. In the case of AF ILC the peak is at (π, π) in all tion corresponds to a physics governed mainly by singlet the range from the isolated ladder, which corresponds to dimers on the ladder rungs and in this case the sign of the gaped “quantum disordered” region, to the isotropic the coupling between these relatively isolated dimers is square lattice, but the extrapolation of its intensity to the irrelevant. On the other hand, for |Jinter | > 0.3, for AF bulk limit becomes nonzero only for Jinter > Jinter,cr , i.e. interladder coupling it is possible that the singlets deloin the “renormalized classical” region.15,14 For FM ILC, calize from a single ladder and finally form a “resonant as shown in Fig. 5(a) for Jinter = −0.2, the situation valence bond” state or that the singlet-triplet excitations is qualitatively different. The peak in S(q) is now at be replaced by gapless magnon excitations. Previous nu(π, π/2), a feature which is similar to the one seen in the merical studies23,14 precisely locate at Jinter ≈ 0.3 the simpler cases of coupled chains and plaquettes. This beposition of the QCP at which the ladder-like spin liquid havior has been found for all clusters considered, and for is replaced by a long range 2D-like AF order thus choosall Jinter < 0 except for finite size effects: the smaller ing the second possibility. The important point we want |Jinter | the larger the size needed to reach the bulk beto suggest is that in both cases the energy of the system havior. This is illustrated for the 4 × 4 cluster. would be lower than for the case of FM ILC where the singlets on the ladders still persist. To illustrate this scenario we have computed on the 16 × 16 cluster the spin0.8 spin correlation functions S(r) = hS0z Srz i for r = (1, 0) (leg direction), (0,1) (rung), and (1,1), inside a ladder, 0.2 (b) (a) and r = (0, 1) between two ladders. These correlations, (x2.5) 0.6 normalized in such a way that S(0) = 1, are shown in absolute value in the inset of Fig. 4 as a function of |Jinter |. NS(q) The differences between FM and AF ILC appear in the (0,1) (rung) correlations, which remain stronger in the 0.4 former case, and most importantly, in the interladder 0.2 2.0 −0.60

0.0

(0,0)

E0

0.6

0.4

0.2

0.0

0.2

−0.70 0.0

0.4 0.2

|Jinter| 0.6 |Jinter | 0.4

0.8 0.6

(π,0)

(0,0)

(π,π)

(π,0)

(0,0)

FIG. 5. Static structure factor obtained by QMC for Jinter = −0.2. (a) For various cluster sizes. The clusters considered are 4 × 4 (circles), 8 × 8 (squares), and 16 × 16 (triangles). The curve with full circles corresponds to the 4 × 4 cluster and Jinter = −0.3. (b) For the 20 × 20 cluster at T = 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.5 and 2.0. The curves have been multiplied by 2.5 for clarity.

S(r) −0.65

(π,π)

0.8

1.0

A highly nontrivial behavior is found if the temperature dependence is analyzed. In Fig. 5(b)) the evolution of the structure factor for the 20×20 cluster and Jinter = −0.2 is shown at T = 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.5 and 2.0. In the high temperature region (T > 0.8) the peak is located at (π, π). As T is lowered the peak starts to shift towards the zero temperature peak (π, π/2) which is reached at T = 0.3. We found almost no variation with cluster size of these two crossover temperatures at this value of Jinter . The fact that only at a finite tempera-

|Jinter|

FIG. 4. Ground state energy per site in the bulk limit of AF (open circles) and FM (full circles) coupled ladders as a function of the absolute value of Jinter . Error bars are smaller than the symbol size. In the inset, the intraladder (1,0) (circles), (0,1) (squares), (1,1) (triangles), and interladder (0,1) (diamonds) spin-spin correlations are shown as a function of |Jinter | for the 16×16 cluster. Open (full) symbols correspond to AF (FM) interladder couplings.

4

of a point analogous to the QCP in the AF ILC case. In the limit of Jinter → −∞ the coupled ladder system becomes equivalent to a system of AF coupled spin-1 chains, where a finite coupling is necessary to change to a gapless regime.24 To answer this question, let us now examine the behavior of the singlet-triplet spin gap as a function of Jinter . Although this is not a convenient quantity to compute with QMC since it involves a difference between absolute values of the energies and then for large clusters the error becomes comparable to its value, we could get an indication of the presence or absence of a gapless region. The gap was computed for finite clusters and then extrapolated to the bulk using the law ∆∞ + b exp(−L/L0 )/L (or a/L2 for the gapless case25 ). The results are depicted in Fig. 6(b). For the AF case, it can be seen a rapid decrease of ∆ as Jinter is increased, confirming earlier predictions and calculations.3,23,14 The gap vanishes at the QCP, Jinter,cr ≈ 0.3. For FM ILC we also obtain a monotonically decreasing behavior, similar to the one found in the previous section for coupled dimers and plaquettes. The gap seems larger to that of AF ILC but it could vanish at Jinter ≈ −0.4 within error bars. The calculation of other quantities like correlation lengths and/or using more powerful techniques should be necessary to obtain a reliable estimation for Jinter,cr .

ture the peak of the magnetic structure factor starts to be incommensurate is reminiscent to the one first found in La1.6−x Nd0.4 Srx CuO4 (x ≈ 0.8) where a charge-stripe order is developed at Tc = 65K followed by a spin-stripe order at a lower temperature Ts = 50K.10 An important quantity to compute is the bulk limit of the peak of the structure factor. This quantity is related with the behavior of spin-spin correlations at the maximum distance along the ladder direction (x-axis), along the direction transversal to the ladders (y-axis) and at the maximum distance of the 2D cluster. In the case of the AF square lattice, this latter quantity is proportional in the bulk limit to the squared staggered magnetization and it should be equal in that limit to the static structure factor at momentum (π, π).26 The finite size scaling of S(π, π/2) is shown in Fig. 6(a) for Jinter = −0.2 and −0.6. We have attempted extrapolations to the bulk limit using both exponential and power laws. Due to the fact that clusters with L = 4, 8, 16 and L = 6, 12 belong to two different sets (which is more noticeable for large values of |Jinter |), the extrapolation procedure is not very reliable. However, as shown in Fig 6(a), one can conclude that S(π, π/2) is zero for Jinter = −0.2 and nonzero for Jinter = −0.6. The finite size behavior of the spin-spin correlation at the maximum distance along the ladder direction, Smax,x , which is the one with smaller errors in our simulations, is similar to the one for S(π, π/2) and the extrapolated values are also zero (nonzero) for Jinter = −0.2 (Jinter = −0.6).

140.0

(b)

(a) 120.0

0.3

(a)

(b)

0.5

0.2

S(q,ω) (a.u.)

0.6

80.0

(0,π)

(0,π)

60.0

0.4 40.0

S(π,π/2)

0.3

(π,0)

(π,0)

∆ 20.0

(π,π)

(π,π) 0.0

0.1

0.0

0.2

0.1

0.0 0.0

Jinter=−0.4

Jinter=0.4

100.0

1/L

0.2

0.0

0.1

0.2

0.3

0.4

2.0

ω/J

3.0

0.0

1.0

2.0

ω/J

3.0

4.0

FIG. 7. Dynamical structure factor obtained by LD on the 4 × 4 cluster vs. frequency at several momenta q for (a) AF and (b) FM interladder couplings. From bottom to top the values of q are (π/2, π/2), (π, π), (π, π/2), (π, 0), (π/2, 0), (π/2, π), (0, π) and (0, π/2).

0

0.1

1.0

0.5

|Jinter|

FIG. 6. (a) S(π, π/2) (open symbols) and Smax,x (full symbols) as a function of 1/L for Jinter = −0.2 (circles) and −0.6 (squares). (b) Singlet-triplet spin gap in the bulk limit of FM (open circles) and AF (full circles) coupled ladders as a function of the absolute value of the interladder coupling constant. In the AF case, the value of Jinter at which ∆ = 0 is taken from Ref. 23. The lines are guides to the eye.

The final part of our study which can eventually lead to a deeper understanding of the excitations involved in this system is the analysis of the dynamical structure factor S(q, ω) which has been done with LD as in the previous section. In this case, we have to limit ourselves to somewhat smaller clusters but we hope that the qualitative features we found will survive in the bulk limit. Results obtained for the 4 × 4 cluster are shown in Fig. 7. For AF ILC (Fig. 7(a)) the peak in S(q, ω) is located at (π, π), as expected in the bulk limit for an AF order. When a FM

This crossover in the behavior of S(π, π/2) as a function of Jinter poses us with the question of the existence

5

ILC is involved (Fig. 7(b)) it can be seen that considerable spectral weight is transferred to the peak at (π, π/2), which becomes also the lowest energy excitation. Results for the 6 × 4 cluster are quite similar and it is quite reassuring that these results are consistent with the ones obtained with QMC and shown in Fig. 5.

at (π, π/2) is reached at a finite temperature and there is a range of temperatures in which an incommensurate peak is present. As mentioned in the previous section, this is reminiscent of the order in which charge and the spin stripes appear in the cuprates as the temperature is decreased. The fact that the spin gap possibly remains finite for somewhat strong values of |Jinter | is also interesting for the stripe scenario of cuprates although in this case our results for the bulk limit are affected by large error bars. Of course, the question of to what extent this model of FM coupled ladders could apply to this scenario should come of detailed comparison with more realistic models like 2D t-J or Hubbard models.

IV. CONCLUSIONS

In summary, we have numerically obtained some exact (except for extrapolation procedures) results for ferromagnetically coupled systems, in particular two-leg ladders. Our main results are embodied in Fig. 5 and Fig. 7, i.e. at zero temperature the peak of the structure factor is located at (π, π/2) and it corresponds to the lowest energy excitation. Fig. 6(a) also suggests finite values of this peak of the structure factor and the spin-spin correlation at the maximum distance along the ladder axis in the bulk limit for strong Jinter . There also two crossover temperatures, a higher one at which the peak starts to shift away from (π, π) and a lower one at which the peak reaches its zero temperature position. Besides the intrinsic interest for the theoretical understanding of spin-1/2 ladder systems, we will try to emphasize in this section their possible relevance for realistic compounds containing ladders. As mentioned in the introduction, we should consider in the first place ladder compounds like SrCu2 O3 27 , Sr14 Cu24 O41 (which upon Ca-doping and under pressure becomes superconducting28 ) and CaV2 O5 . In these compounds, the ladders are coupled forming a trellis lattice. In the former case the interladder couplings are actually ferromagnetic. In the others, due to the frustrated nature of the AF ILC one could speculate that to some extent they could be modeled effectively by FM couplings. For all these compounds, then we predict that neutron scattering experiments would show peaks at (qx , qy ) = (π, π/2), where the x (y) axis is in the direction parallel (perpendicular) to the ladders. As also suggested in the introduction, our results could be related to the striped structure which dynamically appears in the Cu-O planes of high-Tc cuprates. In this case we are able to trace the origin of the neutron scattering peaks observed away from (π, π) to an effective ferromagnetic interaction between π-shifted insulating spin ladders. In fact, this behavior can be observed already for the simplest case of ferromagnetically coupled AF dimers as shown in Section II. In this case, it is easy to verify on small chains by LD or QMC that the peak moves continuously from q = π/2 to π as some of the FM couplings are replaced by AF ones. If this picture could translate to coupled ladders, then one would be lead to the conclusion that some of the spin two-leg ladders are π-shifted while others are in phase in order to reproduce the experimentally observed incommensurate peaks. Another feature we want to emphasize is the temperature evolution of the structure factor (Fig. 5(b)): the peak

ACKNOWLEDGMENTS

We wish to acknowledge many interesting discussions with A. Dobry, A. Greco and A. Trumper. We thank the Supercomputer Computations Research Institute (SCRI) and the Academic Computing and Network Services at Tallahassee (Florida) for allowing us to use their computing facilities.

1

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